An Improvement of Reed’s Treewidth Approximation

Mahdi Belbasi, Martin Fürer

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We present a new approximation algorithm for the treewidth problem which finds an upper bound on the treewidth and constructs a corresponding tree de-composition as well. Our algorithm is a faster variation of Reed’s classical algorithm. For the benefit of the reader, and to be able to compare these two algorithms, we start with a detailed time analysis of Reed’s algorithm. We fill in many details that have been omitted in Reed’s paper. Computing tree decompositions parameterized by the treewidth k is fixed parameter tractable (FPT), meaning that there are algorithms running in time O(f(k)g(n)) where f is a computable function, and g(n) is polynomial in n, where n is the number of vertices. An analysis of Reed’s algorithm shows f(k) = 2O(klogk) and g(n) = n log n for a 5-approximation. Reed simply claims time O(n log n) for bounded k for his constant factor approximation algorithm, but the bound of 2Ω(klogk)n log n is well known. From a practical point of view, we notice that the time of Reed’s algorithm also contains a term of O(k2224kn log n), which for small k is much worse than the asymptotically leading term of 2O(klogk)n log n. We analyze f(k) more precisely, because the purpose of this paper is to improve the running times for all reasonably small values of k. Our algorithm runs in O(f(k)n log n) too, but with a much smaller dependence on k. In our case, f(k) = 2O(k). This algorithm is simple and fast, especially for small values of k. We should mention that Bodlaender et al. [2016] have an algorithm with a linear dependence on n, and Korhonen [2021] obtains the much better approximation ratio of 2, while the current paper achieves a better dependence on k.

Original languageEnglish (US)
Pages (from-to)257-282
Number of pages26
JournalJournal of Graph Algorithms and Applications
Issue number2
StatePublished - 2022

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science
  • Computer Science Applications
  • Geometry and Topology
  • Computational Theory and Mathematics


Dive into the research topics of 'An Improvement of Reed’s Treewidth Approximation'. Together they form a unique fingerprint.

Cite this